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added the example of the profinite completion of the integers to profinite completion of a group.
In the process I noticed that the sub-section structure here and at profinite group had been a bit weird, and I have now tried to clean that up a little.
But the entries still need attention in order to become nice and readable. In particular at profinite completion of a group is plenty of old query-box discussion on how to do it right, Would be nice if somebody with a bit of energy took the query boxes away and edited the entry such as to do it right.
I have cut the query box from profinite completion of a group. The contents are here.
+– {: .query} Tim: Three points: (i) I suggest a hat $\hat{G}$ as notation for completion, (ii) in taking limits you have to specify in what category your are living at that instant! Here it is not the limit in the category of groups that has to be taken but withing the category of topological groups, so you have to consider each finte group as a discrete topological group. Because of this (iii) the actual entry on profinite group takes one to be a pro-object in the category of finite groups. (The way to handle this may need to be discussed to get consensus on which we all like best.)
I am not sure what name the SGA1 type fundamental group should be called. I do not particularly like the one you use below.
David Roberts: This was a knock-together job, I must admit. Actually I think the SGA1 fundamental group is called the algebraic fundamental group - in the case of schemes its also called the etale fundamental group, but I personally think ’algebraic’ would fit better in this example - it is changed!
I’ll also leave off the description of this object to its own page. Sometime I’ll also get around to writing about Grothendieck’s Galois theory.
Mike: Of course, it would be more consistent to use the same definition of profinite group here as at profinite group. As I said there, I prefer the cofiltered-system definition since it is more “basic” and explains why you have to consider topological groups if you want to actually take the limit. Would you object if we defined $\hat{G}$ to be the cofiltered diagram consisting of all finite quotients of $G$, and then observed that, just as always for a profinite group, we could instead take the limit as a topological group?
Tim: That was what I was hinting at in (iii). I think also that the universal property is not yet in its optimal form as it mixes topological ond non-topological things too much. I may try to put together a categorical formulation as a ’gloss’ on this. =– {: .query}
I changed what seemed to be a mistake where it was said that the cone $G\to \hat{G}$ was a morphism in $pro-Fin.Grps$. This would only be the case if $G$ was itself finite.
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